Energy hierarchies governing quarkonium dynamics in Heavy Ion Collisions

In this paper, we critically examine hierarchies between energy scales that determine quarkonium dynamics in the quark gluon plasma. A particularly important role is played by the ratio of the binding energy of species ($E_b$) and the medium scales; temperature ($T$) and Debye mass ($m_D$). It is well known that if these ratios are much larger than one then the dominant process governing quarkonium evolution is dissociation by thermal gluons (gluo-dissociation). On the other hand, if this ratio is much smaller than one then quarkonium dynamics is dominated by screening and Landau damping of the exchanged gluons. Here we show that over most of the evolution, the scale hierarchies do not fall in either limit and one needs to use the full structure of the gluonic spectral function to follow the dynamics of the $Q\bar{Q}$ pair. This has a significant bearing when we follow the quantum dynamics of quarkonia in the medium. The inverse medium relaxation time is also $\sim T$ and if $E_b$ is comparable (or larger) in magnitude to $T$, the quantum evolution of $Q\bar{Q}$ is non-local in time within the Brownian approximation.